The design of the damper is intrinsically linked to the vibrations or oscillations of the mechanical system, where achieving controllable damping is critically significant. To address challenges in tunable dampers, a damper based on annular jet was designed for tensegrity structures. Its key innovation lies in geometric control of damping performance, enabled by a theoretical model correlating geometric parameters to damping behavior through fluid flow within the gap. Subsequently, dynamic modeling and experimental verification were performed for single-degree-of-freedom (DOF) and multi-DOF systems integrated with tensegrity structures. Free vibration experiments and sequential model updating ensured accurate identification of the inherent parameters of the system. A sweep frequency vibration experiment confirmed that the dynamic behavior of the system aligns with the proposed damper design, validating both the damping effect of the annular jet and the modeling accuracy of the tensegrity structure. Next, parametric studies further analyzed the impact of design parameters on damping performance, providing configuration recommendations for damper optimization and potential applications. Furthermore, using the multi-DOF system as an example, the trajectory-based damping performance evaluation demonstrated that the current damper reduces the trajectory envelope by 16.69% compared to the system without it. Additionally, by altering the fluid medium, the proposed damper enables up up to 98.68% reduction in trajectory envelope, underscoring its customizability and significant potential for vibration suppression.